$\newcommand{\fs}{\mathscr{F}}\newcommand{\gs}{\mathscr{G}}$Let ${\bf Sh}(X,{\bf Ring})$ be the category of sheaves of rings on $X$.

Let $\phi:\fs\to\gs$ be a morphism in ${\bf Sh}(X,{\bf Ring})$, and $\operatorname{im}\phi$ the image presheaf of $\phi$. Is it true that if $\phi$ is an epimorhism, then $\gs$ is the sheafification of $\operatorname{im}\phi$ ?

It would be true if we used ${\bf Ab}$ instead of ${\bf Ring}$, because ${\bf Ab}$ is a balanced category.

1 Answer
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This is false. Take $X = \{ \mathrm{p.t.}\}$ so that $\mathsf{Sh}(X,\mathsf{Ring})$ is actually just $\mathsf{Ring}$ and actually the image presheaf is a sheaf. But we know in $\mathsf{Ring}$ that $\mathbb{Z} \rightarrow \mathbb{Q}$ is epic.